# Bounded Variation: Is f:[a,b]-->R Bounded?

• Rasalhague
In summary, the statement "f:[a,b]-->R is of bounded variation" is not equivalent to the statements "f:[a,b]-->R has bounded range" and "f;[a,b]-->R is a bounded function". Being of bounded variation is much stricter than being bounded, as it requires the function to be continuous except in a countable set, have one-sided limits everywhere, and have a derivative almost everywhere.
Rasalhague
Am I right in thinking that the statement "f:[a,b]-->R is of bounded variation" is equivalent to the statements "f:[a,b]-->R has bounded range" and "f;[a,b]-->R is a bounded function".

Hi Rasalhague!

No, you are not correct in thinking that. Being of bounded variation is much more strict than being bounded. For example, the function on [0,1]

$$f(x)=\sin(1/x)$$

is bounded between -1 and 1, but it is not of bounded varietion. Another example is the Dirichlet function on [0,1]:

$$f(x)=\left\{\begin{array}{c} 1~\text{if}~x\in \mathbb{Q}\\ 0~\text{if}~x\notin \mathbb{Q}\\ \end{array}\right.$$

This is bounded, but not of bounded variation. It can be shown that the following must hold for functions of bounded variation:

• The function is continuous everywhere except possibly in a countable set.
• The function has one-sided limits everywhere.
• The function has a derivative almost everywhere (i.e. except in a set of measure 0).

so you see that being of bounded variation is pretty strict.

Argh, thanks micromass, I see now where my confusion came from. For some reason my mind was just blanking out the summation sign! Oopsh. Ack. Blushy-faced emoticon. I think it's asking for a rest :)

## What is bounded variation?

Bounded variation is a mathematical concept that measures the "smoothness" of a function. It looks at the total change in a function over a specific interval and determines whether it is finite or infinite.

## How is bounded variation calculated?

To calculate bounded variation, we take the absolute value of the differences between consecutive function values and add them together. If this value is finite, the function is said to have bounded variation.

## What does it mean for a function to be bounded?

A function is considered bounded if its values do not exceed a certain bound or limit. This means that the function does not have any extreme or infinite values, and its behavior is predictable within a certain range.

## Can a function have bounded variation but not be bounded?

Yes, a function can have bounded variation but not be bounded. Bounded variation only looks at the total change in a function over an interval, while boundedness looks at the values of the function itself. A function can have a finite total change, but its values can still exceed a certain bound, making it not bounded.

## Why is bounded variation important?

Bounded variation is important in many areas of mathematics, including calculus, analysis, and differential equations. It helps us understand the behavior and properties of functions, such as continuity, differentiability, and integrability.

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